gasket (Example 2.1) and nested fractals (e.g. Example 2.3) introduced by ... Sierpinski gasket which contains the Brownian motion as a special case. In.
765
J. Math. Kyoto Univ. (JMKYAZ) 33-3 (1993) 765-786
Regularity, closedness and spectral dimensions
of the Dirichlet forms on P.C.F. self-similar sets By Takashi KUMAGAI
§ 1. Introduction Today, there has been many works about the self-similar sets. O ne well known framework for describing them is due to Hutchinson ([21): Let F1(1 i K such that 7ro d s=Fs.7r f o r every s E S . Further, f o r coES2n, w e denote F.= F.,0FOE, 0 •-•.F., and I n particlar, K s— Fs(K ) f o r sE S . 2
Remark that Hutchinson's self-similar set is a self-similar stucture in the sense of Definition 2.1 by taking 7r(co)= Definition 2.2. Let ± = ( K , S , { F } Then the critical set of .1' is defined by
(2.1)
ses)
be a self-similar structure on K.
C(-C)=7r-'(U s,tes,s*t(K , fl Kt))
is defined by
and the post critical set of
(2.2)
P ( ± ) =U n ic in (C (± )) .
± is called post critically finite, or P.C.F. for short, if P P ( . . f ) is a finite set.
In the following, we consider a P. C. F. self-similar set (K , S, {F5}55). In order to m ake our fram ew ork clear, we give some examples.
Example 2.1: Sierpinski Gasket
S={1, 2, 3) . 7r(C)={ qi, q2, q3}; '( q ) = { 1 ,
3 ) , R- - 1 (q2)= M , 31},
(q 1) =
21) .
7r(P)={ ti, P2, P3); 7 (P O - { ; -1
} f o r i =1 , 2, 3 .
Example 2.2: Hata's Tree-like Set
S=11, 2) .
7r(C)=1q1; 7r (q1)={112, 21), -1
7r(P)= {p i, p 2 , p3); 71- i(p = { ; } f o r i E S,'( p 3 ) = { 1 } , -
Example 2.3: LindstrOm's snowflake
S=11, 2, 3, 4, 5, 6, 7) . 7 r(C )={ q 1 : 1 i .
12} ;
7r i(q ,)= ti ,
1 for
-
71 (q,)= { 7 < b, < i x i 3 > --1
.
}
f o r 7 0 . T hen ( 9 , E
> 1
•
Also, from the above lemma, there is a positive constant C such that c(aA)
-
- e (g, g) L li(g)(x (w ))—
f
PCN (i)••• a( i) s2
g(R - ([0), qU)].))1 2 (da))
Ii ( g ) (
( ))—g(71 ( Ew, qU)1m))1 -
2
- -
(dco)
e times
= Opcnn.n . Lli(g)(7rap(j), adm))— g(z(P(i)))1 2 ri(da)) .
(Here m is chosen as s + nt have i
i
—
m.) Therefore, using the triangle inequality, we
19(71-(P(j)))1— (Lli(g)(7r([p(j), w]m))1 f) (d(0))
(3.4)
2
-
2 , (d (0 )) 4-/ in the operator norm . A s Tm is a compact operator, we see / is a compact operator. Thus /* is also a compact operator. As /* equals to the a-order resolvent operator, the a-order resolvent is compact.
776
T K umagai
Remark 3 .6 . Given positive numbers pi,
fiN such that E;T=iti.,=1,
co (thus / ->co), we finally obtain 0 0 . Thus we obtain the existence of the
harmonic structure in this c a s e . W e give one typical example which is weakly symmetric. By the above results, we have non-degenerate Dirichlet forms on this fractal.
Example 7.1 S={1, 2, 3, 4, 5, 6) . 71- ( C )= Iq i:1 < i< 1 0 1 ; ir - 1 (q i)= {i< i:k 1 > ,< i+ 1 > i}
for 1 i 5,
T. Kum agai
786 Example 7.1
0.. V„ - - - - - - - - - ,4 - - • R (qi)={6< i >, < i>}for 6_.
